Problem 53 - Applied Calculus

Question

Show that \(y-y_{1}=m\left(x-x_{1}\right)\) simplifies to \(y=m x+b\) if the point \(\left(x_{1}, y_{1}\right)\) is the \(y\) -intercept \((0, b)\)

Step-by-Step Solution

Answer

The equation simplifies to \(y = mx + b\) when \((x_1, y_1)\) is the \(y\)-intercept \((0, b)\).

Step 1: Substitute the Point into the Equation

First, substitute the coordinates of the point \((x_1, y_1)\), which is given as the \(y\)-intercept \((0, b)\), into the equation \(y - y_1 = m(x - x_1)\). This gives you:\[y - b = m(x - 0)\]

Step 2: Simplify the Expression

Simplify the expression from Step 1. The equation \(y - b = m(x - 0)\) simplifies to:\[y - b = mx\]

Step 3: Solve for y

To isolate \(y\), add \(b\) to both sides of the equation:\[y = mx + b\]

Step 4: Conclusion: Simplification Complete

The equation \(y - y_1 = m(x - x_1)\), after substituting the point \((0, b)\) and simplifying, results in the equation \(y = mx + b\). This shows that the simplified form is correct when the point is the \(y\)-intercept.

Key Concepts

Slope-Intercept Form

When talking about linear equations, the slope-intercept form is one of the most common ways to express them. It's written as \( y = mx + b \). In this equation:

• \(m\) represents the slope of the line
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis

Using this form allows you to easily identify both the slope and y-intercept, so you can graph the line quickly.

In the equation, the slope \(m\) indicates how steep the line is. A larger value for \(m\) means a steeper slope. If the slope is positive, the line goes up from left to right, and if it's negative, the line goes down. The y-intercept \(b\) shows the starting point of the line on the y-axis, giving the height where the line begins from the y-axis. Understanding the slope-intercept form helps in graphing linear equations and in predicting how changes in the equation affect the line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses coordinates to represent geometric shapes and solve geometric problems. In the context of linear equations, it focuses on using both x and y coordinates to describe lines.

Every point on a line in coordinate geometry is represented by an ordered pair \((x, y)\). A key component here is knowing two main components:

• The slope, which measures the angle or steepness of the line
• The y-intercept, indicating the point where the line meets the y-axis

Coordinate geometry allows us to use equations, like the slope-intercept form, to not only draw lines but to determine relationships between different lines or points.

By understanding the position of points and their relationship to each other using this coordinate system, you can solve problems involving distance, midpoint, and line intersections efficiently. It provides a graphical understanding of the algebraic processes behind linear equations.

Simplification Steps

Simplifying an equation involves breaking it down to its simplest form while retaining the same solutions. In our example, the equation \( y - y_1 = m(x - x_1) \) needed to be simplified to \( y = mx + b \), assuming \( (x_1, y_1) \) is the y-intercept \((0, b)\).

Here’s how simplification works:

• **Substitution**: Replace variables with the given coordinates, replacing \((x_1, y_1)\) with \((0, b)\).
• **Simplification**: In the expression \( y - b = m(x - 0) \), we rewrite it as \( y - b = mx \).
• **Solve for \(y\)**: Add \(b\) to both sides to isolate \(y\), giving us \( y = mx + b \).

Each step streamlines the equation to its simplest, most practical format. Simplification helps in providing a clear and direct form of the equation, making it easier to interpret and use in further calculations.